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From: vnuri@eecs.wsu.edu (Veyis Nuri - EECS)
Subject: Re: Question: Wavelet Transform vs FFT
Message-ID: <CyrDDv.ItB@serval.net.wsu.edu>
Sender: news@serval.net.wsu.edu (Greg Bell)
Organization: School of EECS, Washington State University
References: <395ah7$1vh@wynkyn.ecs.soton.ac.uk> <AZ.94Nov3100521@saturn.analog.com>
Date: Fri, 4 Nov 1994 19:41:53 GMT
Lines: 49

In article <AZ.94Nov3100521@saturn.analog.com>, az@saturn.analog.com (Alex Zatsman) writes:
|> In article <395ah7$1vh@wynkyn.ecs.soton.ac.uk> jk94r@ecs.soton.ac.uk (Joseph Kuan) writes:
|> 
|> > 	   Wavelet transform is orthogonal, (FFT as well) but what is the
|> >    meaning of orthogonal and what is it related to the image
|> >    processing?
|> 
|> 1. Orthogonality of the wavelet basis  means that that decomposition
|> and reconstruction  is done using the same base functions. 
|> 

Not exactly, they are related to each other.

|> 2. As  someone mentioned in an earlier  thread on wavelets --  sorry I
|> don't remember  the  name --  orghogonality  also  implies  Parseval's
|> equation, i.e. the energy of the signal is equal to the sum of squares
|> of its coefficients in the orghogonal basis.

This may be of interest but not much crucial. Necessary justifications
can be made to find the relation between the source energy and the 
energy of wavelet coefficients (subbands) for non-orthogonal bases
(non-paraunitary filter banks).

In addition to above features, orthogonal bases (para-unitary filter
banks) imply that the output of each wavelet basis (subband) 
is uncorrelated with the others (do not confuse this with 
intra-band correlation). This significantly reduces complexity of 
derivations in compression of wavelet coefficients (called subband coding). 
However, as I mentioned before, this is not an crucial issue and modifications 
can be made.

Overall, since there has been a great amount of research done 
on orthogonal functions, this makes it easy to adopt available results to wavelet
(filter bank) applications if orthogonal wavelets (para-unitary filter banks)
are used.

But, non-orthogonal wavelets (non-para-unitary filter banks) carry some 
desirable features (giving up the orthogonality condition), especially
in image processing applications, which their orthogonal
(para-unitary) counterparts lack. So, I suggest you broaden your research
including non-orthogonal wavelets (non-paraunitary filter banks). One more
point, even though wavelets and filter banks are discussed together
in the literature, filter banks are a bigger class than wavelets and
some filter banks which are even not considered to be a wavelet
may be more suitable for your application.

Have a nice day,

-veyis
